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sagemanifolds
GitHub Repository: sagemanifolds/IntroToManifolds
Path: blob/main/06Manifold_VectSpaces_Modules.ipynb
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Kernel: SageMath 9.6

6. The notion of module

This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).

version()
'SageMath version 9.6, Release Date: 2022-05-15'

If F is a field, a set V is called a vector space over F if there is an operation of addition  (x,y)x+y  \ (x,y) → x+y\ \ from V×V to V, and a scalar multiplication operation   (α,x)αx  \ \ (α,x) → αx\ \ from F×V to V, such that the following properties are satisfied.
(i) (u+v) + w = u + (v+w) for all u, v, w ∈ V,
(ii) u+v=v+u for all u, v ∈ V,
(iii) There exists an element 0 ∈ V such that 0+v=v+0 for all v ∈ V,
(iv) For each v ∈ V there exists u ∈ V such that v+u=0,
(v) 1v=v for all v ∈ V,
(vi) (αβ)v=α(βv)(\alpha\beta)v=\alpha(\beta v) for all α,βF\alpha,\beta\in F and v∈V,
(vii) (α+β)v=αv+βv(\alpha+\beta)v=\alpha v+\beta v for all α,βF\alpha,\beta\in F and v∈V,
(viii) α(u+v)=αu+αv\alpha(u+v)=\alpha u+\alpha v for all αF\alpha\in F and uu∈V.

The formal definition of the module M over the ring R is exactly as above (with V replaced by M and F replaced by R) but we relax the requirement that F be a field, and instead allow an arbitrary ring R (with unity). We shall restrict ourselves to the commutative rings R.

A free module of finite rank over a commutative ring R is a module MM over R that admits a finite basis, that is a finite family {e1,,ek}\{e_1,\ldots,e_k\}, which spans MM, i.e., for every vMv\in M,  v=i=1kaiei\ v=\sum_{i=1}^k a_ie_i for some ai a_i\in R, and eie_i are linearly independent i.e., i=1kaiei=0\sum_{i=1}^k a_ie_i=0 implies that all aia_i are zero.

Since R is commutative, it has the invariant basis number property, so that the rank (dimension) of the free module M is defined uniquely, as the cardinality of any basis of MM.

General remark on free modules in SageMath

Basic motivation for introducing free modules into consideration in SageMath Manifolds is the fact that the sets of vector fields and tensor fields on a parallelizable open subset U of the manifold, are free modules over the ring of scalar fields on U.

Some frequently used commands from SageMath FiniteRankFreeModule:

an_element endomorphism sym_bilinear_form automorphism hom tensor bases irange tensor_from_comp basis linear_form tensor_module change_of_basis rank alternating_form default_basis set_change_of_basis dual_exterior_power dual set_default_basis exterior_power

In our examples we will use mainly the SageMath symbolic ring SR.


Example 6.1

FiniteRankFreeModule in SageMath:

Symbolic ring SR is considered as a field:

SR.is_field() # Symbolic ring is a field
True

so the finite rank free modules over SR are in fact vector spaces:

V = FiniteRankFreeModule(SR,4,name='V') # rank 4 free module V over SR #V? --detailed explanations concerning FiniteRankFreeModule V # information on V
4-dimensional vector space V over the Symbolic Ring

Defining V as a finite rank free module over a field we obtain a vector space without predefined basis.

V.bases() # check that V has no predefined basis
[]

Later we shall see that without any specific coordinate choice, no basis can be distinguished in a tangent space. This is one more motivation for using modules in SageMath Manifolds.


Let us check that defining vector space with VectorSpace command, we introduce a predefined basis.

W = VectorSpace(SR,4); W
Vector space of dimension 4 over Symbolic Ring
W.basis()
[ (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) ]

In SageMath the VectorSpace of rank 4 over R is in fact R^4 -the Cartesian power of R.

W == SR^4
True

but an analogous FiniteRankFreeModule is not:

V == SR^4
False

Example 6.2

To define a vector in SageMath Manifolds we need some basis.

%display latex M = FiniteRankFreeModule(SR, 4, name='M') # module M e = M.basis('e') # basis of M # e? -detailed explanations concerning bases # range(1,5)=[1,2,3,4] --coefficients of the linear combination u = M(range(1,5),name='u');u.disp() # show vector

u=e0+2e1+3e2+4e3\displaystyle u = e_{0} + 2 e_{1} + 3 e_{2} + 4 e_{3}

Vectors can be defined also as linear combinations of the basis.

al = srange(1,5) # a more flexible version of range # linear combination of basis: w = sum([al[k]*e[k] for k in range(4)]) w.disp() # show w

e0+2e1+3e2+4e3\displaystyle e_{0} + 2 e_{1} + 3 e_{2} + 4 e_{3}

# a more general linear combination of a basis # \alpha+[Tab] gives the Greek letter alpha al = var('α',n=4) # α_i, i=1,2,3,4 # linear combination of basis: w = sum([al[k]*e[k] for k in range(4)]) w.disp() # show w

α0e0+α1e1+α2e2+α3e3\displaystyle α_{0} e_{0} + α_{1} e_{1} + α_{2} e_{2} + α_{3} e_{3}


Linear transformations in finite rank free module


By the linearity of the map Φ:MN\Phi:M\to N between two modules over RR we mean the condition

Φ(av+bw)=aΦ(v)+bΦ(w),forv,wM,a,bR.\Phi (av + bw ) = a\Phi (v) + b\Phi (w ),\quad\text{for}\quad v,w\in M,\quad a,b\in R.

Automorphism


Automorphism of the module M is a bijective linear transformation M \to M.

An automorphism allows for example for defining a new basis.
If more than one bases are defined, the first one is the default one.


Example 6.3

Let us define a module with two bases and an automorphism defining the change of basis.

%display latex M = FiniteRankFreeModule(SR, 4, name='M') # 4-dim module over SR a = M.automorphism() # automorphism of M # a? --detailed explanations concerning module automorphism e = M.basis('e') # define default basis f = M.basis('f') # define second basis # diagonal matrix with diagonal (4,3,2,1) a[e,:]=diagonal_matrix(4, srange(4,0,-1)); # matrix of automorphism M.set_change_of_basis(e, f, a) # define change of basis u = M(range(1,5),name='u') # define vector # displaying the vector in non-default basis needs # the basis name as an argument u.disp(),'________',u.disp(f) # vector u in bases e and f

(u=e0+2e1+3e2+4e3,________,u=14f0+23f1+32f2+4f3)\displaystyle \left(u = e_{0} + 2 e_{1} + 3 e_{2} + 4 e_{3}, \verb|________|, u = \frac{1}{4} f_{0} + \frac{2}{3} f_{1} + \frac{3}{2} f_{2} + 4 f_{3}\right)

M.default_basis() # default basis

(e0,e1,e2,e3)\displaystyle \left(e_{0},e_{1},e_{2},e_{3}\right)

Matrix of the automorphism:

a.matrix() # matrix of automorphism a

(4000030000200001)\displaystyle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)


Example 6.4

The new basis can be also defined component by component.

%display latex M=FiniteRankFreeModule(SR,4,name='M') # 4-dim module over SR e = M.basis('e') # basis e f = M.basis('f', from_family= # new basis f contains lin.comb. (4*e[0],3*e[1],2*e[2],e[3])) # of elements of e u = M(range(1,5),name='u'); # vector u u.disp(),'______',u.disp(f) # u in bases e and f

(u=e0+2e1+3e2+4e3,______,u=14f0+23f1+32f2+4f3)\displaystyle \left(u = e_{0} + 2 e_{1} + 3 e_{2} + 4 e_{3}, \verb|______|, u = \frac{1}{4} f_{0} + \frac{2}{3} f_{1} + \frac{3}{2} f_{2} + 4 f_{3}\right)

Matrix of the change of basis e \to f:

M.change_of_basis(e,f).matrix(e) # matrix of change of basis e->f

(4000030000200001)\displaystyle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)

Matrix of the change of basis f \to e:

M.change_of_basis(f,e).matrix(e) # matrix of change of basis f->e

(1400001300001200001)\displaystyle \left(\begin{array}{rrrr} \frac{1}{4} & 0 & 0 & 0 \\ 0 & \frac{1}{3} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)


Endomorphism


Endomorphism of the module M is a general linear transformation M \to M.


Example 6.5

Define an endomorphism using arbitrarily chosen matrix.

M = FiniteRankFreeModule(SR,4,name='M') # 4-dim module over SR e = M.basis('e') # basis e ma = matrix(4,4,range(16)) # matrix of endomorphism phi = M.endomorphism(ma, basis=e, # define an endomorphism name='phi',latex_name=r'\Phi') # Phi u = M(range(1,5),name='u'); # vector u phi(u).disp() # Phi(u)

Φ(u)=20e0+60e1+100e2+140e3\displaystyle \Phi\left(u\right) = 20 e_{0} + 60 e_{1} + 100 e_{2} + 140 e_{3}


Using general symbolic matrices


Example 6.6

To obtain general N×NN\times N matrices with elements aija_{ij} we can use for example the following definition:

aa = [[var('a'+str(i)+str(j)) # 4x4 symbolic matrix is for j in range(4)] # defined by list of lists for i in range(4)]
aa # list of lists (list of rows)

[[a00,a01,a02,a03],[a10,a11,a12,a13],[a20,a21,a22,a23],[a30,a31,a32,a33]]\displaystyle \left[\left[a_{00}, a_{01}, a_{02}, a_{03}\right], \left[a_{10}, a_{11}, a_{12}, a_{13}\right], \left[a_{20}, a_{21}, a_{22}, a_{23}\right], \left[a_{30}, a_{31}, a_{32}, a_{33}\right]\right]

General endomorphism of a 4-dimensional module:

M = FiniteRankFreeModule(SR,4,name='M') # 4-dim module M over SR e = M.basis('e') # basis of M phi = M.endomorphism(aa, basis=e, # define an endomorphism name='phi',latex_name=r'\Phi') # using matrix for k in range(4): # show the values Phi(e_i) show(phi(e[k]).disp()) # coefficients of phi(e_k) are taken from # the k-th column of phi matrix

Φ(e0)=a00e0+a10e1+a20e2+a30e3\displaystyle \Phi\left(e_{0}\right) = a_{00} e_{0} + a_{10} e_{1} + a_{20} e_{2} + a_{30} e_{3}

Φ(e1)=a01e0+a11e1+a21e2+a31e3\displaystyle \Phi\left(e_{1}\right) = a_{01} e_{0} + a_{11} e_{1} + a_{21} e_{2} + a_{31} e_{3}

Φ(e2)=a02e0+a12e1+a22e2+a32e3\displaystyle \Phi\left(e_{2}\right) = a_{02} e_{0} + a_{12} e_{1} + a_{22} e_{2} + a_{32} e_{3}

Φ(e3)=a03e0+a13e1+a23e2+a33e3\displaystyle \Phi\left(e_{3}\right) = a_{03} e_{0} + a_{13} e_{1} + a_{23} e_{2} + a_{33} e_{3}

%display latex # matrix of endomorphism phi.matrix()

(a00a01a02a03a10a11a12a13a20a21a22a23a30a31a32a33)\displaystyle \left(\begin{array}{rrrr} a_{00} & a_{01} & a_{02} & a_{03} \\ a_{10} & a_{11} & a_{12} & a_{13} \\ a_{20} & a_{21} & a_{22} & a_{23} \\ a_{30} & a_{31} & a_{32} & a_{33} \end{array}\right)


Homomorphisms of modules


The general linear maps between modules M, N are homomorphisms.

A bijective homomorphism is called isomorphism.


Example 6.7

Define two modules and a homomorphism between them.

M = FiniteRankFreeModule(SR,4,name='M') # 4-dim module M over SR N = FiniteRankFreeModule(SR,2,name='N') # 2-dim module over SR e = M.basis('e') # basis of M f = N.basis('f') # basis of N ma = matrix(2,4,lambda i,j:i+j) # matrix a_ij=i+j phi = M.hom(N,ma); phi # define homomorphism # with matrix ma

Generic morphism: From: 4-dimensional vector space M over the Symbolic Ring To: 2-dimensional vector space N over the Symbolic Ring\displaystyle \mbox{Generic morphism: From: 4-dimensional vector space M over the Symbolic Ring To: 2-dimensional vector space N over the Symbolic Ring}

# mathematical object of which "phi" is an element phi.parent()

Hom(M,N)\displaystyle \mathrm{Hom}\left(M,N\right)

Let us check the additivity property using general vectors V, W.

v = var('v',n=4) #(v0,v1,v2,v3) # components of vector V w = var('w',n=4) #(w0,w1,w2,w3) # components of vector W V = M(list(v)) # vector V W = M(list(w)) # vector W phi(V+W) == phi(V) + phi(W) # check the additivity

True\displaystyle \mathrm{True}


The value of the homomorphism phi on the vector V in the basis of N:

phi(V).disp() # phi(V)

(v1+2v2+3v3)f0+(v0+2v1+3v2+4v3)f1\displaystyle \left( v_{1} + 2 \, v_{2} + 3 \, v_{3} \right) f_{0} + \left( v_{0} + 2 \, v_{1} + 3 \, v_{2} + 4 \, v_{3} \right) f_{1}

The matrix of the homomorphism phi:

phi.matrix() # matrix of homomorphism

(01231234)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \end{array}\right)


Linear forms on modules. Dual module


By a linear form or a linear functional on a module  M \ M\ over RR we mean the map f:MRf:M\to R such that

f(av+bw)=af(v)+bf(w),forv,wM,a,bR.f (av + bw ) = af(v) + bf(w ),\quad\text{for}\quad v,w\in M,\quad a,b\in R.

The dual module MM^* is the module of linear forms on a module MM.


Example 6.8

The module of all linear functionals or linear forms on M in SageMath Manifold is M.dual():

M = FiniteRankFreeModule(SR,4,name='M') # 4-dim module over SR Mstar = M.dual(); Mstar # Since SR is a field, M is a vector space

M\displaystyle M^*


Dual basis


SageMath dual.basis allows to define a dual basis to a given basis  {ei}i=1n\ \{e_i\}_{i=1}^n, i.e., the set of linear functionals  {ei}i=1n \ \{e^i\}_{i=1}^n\ such that ei(ej)=δji.\quad e^i(e_j)=\delta^i_j.


Example 6.9

Define dual basis in SageMath Manifolds.

# continuation e = M.basis('e'); # basis of M ep = e.dual_basis(); ep # dual basis

(e0,e1,e2,e3)\displaystyle \left(e^{0},e^{1},e^{2},e^{3}\right)

Let us check that ei(ej)=δji\quad e^i(e_j)=\delta^i_j:

%display latex matrix(4,4,lambda i,j:ep[i](e[j])) # matrix e^i(e_j)

(1000010000100001)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)


Linear forms on a module M are linear combinations of elements of the dual basis.


Example 6.10

Define a linear form in SageMath Manifolds.

# continuation a = M.linear_form('a') # linear form a on M a[:] = 1,2,3,4 # define components of a a.disp() # show a

a=e0+2e1+3e2+4e3\displaystyle a = e^{0} + 2 e^{1} + 3 e^{2} + 4 e^{3}

Linear forms are elements of the dual module MM^*.

# mathematical object of which "a" is an element print(a.parent())
Dual of the 4-dimensional vector space M over the Symbolic Ring
Mstar = M.dual() a.parent() == Mstar # check if parent of a is M dual

True\displaystyle \mathrm{True}

What's next?

Take a look at the notebook Smooth functions and pullbacks.